Computational Methods for Calculating Accurate Enthalpies of

Chapter 10

Downloaded by NORTH CAROLINA STATE UNIV on September 9, 2013 | http://pubs.acs.org Publication Date: February 1, 1998 | doi: 10.1021/bk-1998-0677.ch010

Computational Methods for Calculating Accurate Enthalpies of Formation, Ionization Potentials, and Electron Affinities 1

2

Larry A . Curtiss and Krishnan Raghavachari 1

Chemical Technology Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439 Bell Laboratories, Lucent Technologies, Murray Hill, N J 07974 2

In this chapter we describe two methods being used in computational thermochemistry. The first is Gaussian-2 theory, which is based on a sequence of well-defined ab initio molecular orbital calculations. It has proven to be a useful technique for the calculation of accurate bond energies, enthalpies of formation, ionization potentials, electron affinities, and proton affinities. The second is density functional theory which is developing as a cost-effective procedure for studying molecular properties and energies. Quantum chemical methods for the calculation of thermochemical data have developed beyond the level of just reproducing experimental data and can now make accurate predictions where the experimental data are unknown or uncertain. In this chapter we review two theoretical methods that are being used in computational thermochemistry. The more accurate of these methods is Gaussian-2 (G2) theory (7). It was the second in a series of methods termed Gaussian-n theories (1-3) based on ab initio molecular orbital theory for the calculation of energies of molecular systems containing the elements H - C l . The other method is density functional theory ( D F T ) which is less accurate, but is a cost-effective method for including correlation effects that are needed for thermochemical calculations. In this chapter we first review the elements of Gaussian-2 theory and the test set of 125 reaction energies that was developed to assess its reliability. This is followed by a description of several variants of G 2 theory that have been proposed for saving computational time or improving accuracy and the extension of the method to include third-row non-transition metal elements. Then we provide a brief overview of density functional theory and several functionals currently being used in computational quantum chemistry. Recently, the original test set of 125 energies has been expanded to include larger and more diverse molecules. This new test set has been used to critically evaluate G 2 theory, some of its variants, and several D F T

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© 1998 American Chemical Society

In Computational Thermochemistry; Irikura, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1905.

10.

CURTISS & R A G H A V A C H A R I

Computation of Enthalpies of Formation

111

methods. The results of this assessment are described. Finally, examples of several applications of G 2 theory are discussed.


Theoretical Methods Gaussian-2 Theory. Gaussian-2 theory is a composite technique in which a sequence of well-defined ab initio molecular orbital calculations is performed to arrive at a total energy of a given molecular species. Geometries are determined using second-order Moller-Plesset perturbation theory. Correlation level calculations are done using Moller-Plesset perturbation theory up to fourth-order and with quadratic configuration interaction. Large basis sets including multiple sets of polarization functions are used in the correlation calculations. A series of additivity approximations makes the technique fairly widely applicable. Unlike many other approaches, it is not dependent on calibration with experimental data for related species either through isodesmic reactions or in some other manner. It does have a single molecule-independent semiempirical parameter which is chosen by fitting to a set of accurate experimental data. The principal steps in G 2 theory are as follows. 1. A n initial equilibrium structure is obtained at the Hartree-Fock (HF) level with the 6-31G(d) basis (4). Spin-restricted ( R H F ) theory is used for singlet states and spinunrestricted Hartree-Fock theory ( U H F ) for others. 2. The HF/6-31G(d) equilibrium structure is used to calculate harmonic frequencies, which are then scaled by a factor of 0.8929 to take account of known deficiencies at this level (5). These frequencies give the zero-point energy A ( Z P E ) used to obtain E in step 7.

0

3. The equilibrium geometry is refined at the MP2/6-31G(d) level [Moller-Plesset perturbation theory to second order with the 6-31G(d) basis set] using all electrons for the calculation of correlation energies. This is the final equilibrium geometry in the theory and is used for all single-point calculations at higher levels of theory in step 4. A l l of these subsequent calculations include only valence electrons in the treatment of electron correlation. 4. The first higher level calculation is at full fourth-order Moller-Plesset perturbation theory (4) with the 6-31 lG(d,p) basis set, i.e., MP4/6-31 lG(d,p). This energy is then modified by a series of corrections from additional calculations including (a) a correction for diffuse functions (4), AE(+); (b) a correction for higher polarization functions (4) on non-hydrogen atoms, AE(2df); (c) a correction for correlation effects beyond fourth-order perturbation theory using the method of quadratic configuration interaction (6), AE(QCI); and (d) a correction for larger basis set effects and for nonadditivity caused by the assumption of separate basis set extensions for diffuse functions and higher polarization functions, AE(+3df,2p). The single-point energy calculations required for these corrections are described in Table I.


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Table I. Energy corrections for G2, G2(MP2), and G2(MP2,SVP) theory Method Step Corrections G2

4(a)

AE(+) = E[MP4/6-311 +G(d,p)] -E[MP4/6-311G(d,p)] AE(2dO = E[MP4/6-311 G(2df ,p)] -E[MP4/6-311G(d,p)] AE(QCI) = E[QCISD(T)/6-311 G(d,p)]

4(b)


4(c)

-E[MP4/6-311G(d,p)] AE(+3df ,2p) = E[MP2/6-311 +G(3df ,2p)] -E[MP2/6-311G(2df,p)] -E[MP2/6-311+G(d,p)] +E[MP2/6-311G(d,p)]

4(d)

G2(MP2)

4'(d)

G2(MP2,S VP)

4 (c)

A E ' (+3df ,2p) = [E(MP2/6-311 +G(3df,2p)] [E(MP2/6-311G(d,p)]

M

AE"(QCI) = E[QCISD(T)/6-31 G(d)] -E[MP4/6-31G(d)] AE"(+3df,2p) = [E(MP2/6-311+G(3df,2p)] - [E(MP2/6-31G(d)]

M

4 (d)

5. The MP4/6-31 lG(d,p) energy and the four corrections from step 4 are combined in an additive manner: E(combined) = E[MP4/6-311 G(d,p)] + AE(+) + AE(2df) + A E ( Q C I )

(1)

+ AE(+3df,2p) 6. A "higher level correction" ( H L C ) is added to take into account remaining deficiencies E ( G 2 ) = E(combined) + A E ( H L C ) .

(2)

e

The H L C is equal to - A n - Bn ,where the n and n are the number of p and a valence electrons, respectively, with n > n . For G 2 theory, A = 4.81 mhartrees and B = 0.19 mhartrees (equivalent to 5.00 mhartrees per electron pair). The B value was chosen so that E is exact for the hydrogen atom (2). The A value was determined to give a zero mean deviation from experiment for the atomization energies of 55 molecules having well-established experimental values. The higher level correction makes G 2 theory "semi-empirical," though only a single molecule-independent parameter is used. p

a

p

a

a

p

e


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7. Finally, the total energy at 0 K is obtained by adding the zero-point correction, obtained from the frequencies of step 2 to the total energy: E (G2)=E (G2) + A(ZPE) 0

(3)

e

The energy E is referred to as the " G 2 energy."


0

Gaussian-2 theory was tested on a total of 125 reaction energies (55 atomization energies, 38 ionization energies, 25 electron affinities, and 7 proton affinities), chosen because they have well-established experimental values (1-3). The molecules in this test set contain elements from the first- and second-rows of the periodic chart. The mean absolute deviation of G 2 theory for this test set from experiment is 1.21 kcal/mol, and the maximum deviation is for the atomization energy of S 0 (-5.0 kcal/mol). The average absolute deviations of G 2 theory from experiment for the different types of reaction energies in this test set are listed in Table II. The set of 125 reaction energies has since been used by others to test new quantum chemical methods and is often referred to as the " G 2 test set." 2

Table II. Average absolute deviation (in kcal/mol) between theory and 5

a

b

Reaction T y p e Method All EA PA AE IP G2 1.04 1.21 1.29 1.24 1.16 G2(MP2) 0.64 1.58 1.99 1.32 1.86 G2(MP2,SVP) 2.08 0.81 1.63 1.34 1.93 SVWN 25.96 17.21 5.85 15.70 39.59 BLYP 2.42 1.80 4.49 4.75 5.99 BPW91 2.22 1.53 4.48 5.32 5.29 BP86 7.31 1.51 5.34 4.91 10.51 B3LYP 2.81 2.49 1.29 3.88 2.40 B3PW91 2.43 1.17 2.89 2.59 3.96 B3P86 13.77 10.69 1.09 7.84 14.57 G 2 results from Ref. 1; G2(MP2) results from Ref. 9; G2(MP2,SVP) results from Ref. 11. The DFT results (unpublished work) are based on the 6-311+G(3df,2p) basis set, MP2(full)/6-31G(d) geometries, and scaled (0.893) HF/6-31G(d) frequencies. A E = atomization energy (55); IP = ionization potential (38); E A = electron affinity (25); P A = proton affinity (7); number of molecules in parentheses.

The final total energy is effectively at the QCISD(T)/6-311+G(3df,2p) level if the different additivity approximations work well. The validity of these additivity approximations was investigated by performing complete QCISD(T)/6311+G(3df,2p) calculations on the set of 125 test reactions, and in most cases, the additivity approximations were found to work well (7). A l l of the calculations required for G 2 theory are available in the quantum chemical computer program Gaussian 94 (8).


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The steps in G l theory (2,3), the predecessor to G 2 theory, are the same as in G 2 theory except step 4(d) is not included and the value of A in the H L C is 5.95 mhartrees instead of 4.81 mhartrees. The final energy in G l theory is effectively at the QCISD(T)/6-311+G(2df,p) level and is less accurate than G 2 theory (average absolute deviation of 1.58 kcal/mol for the 125 reaction energies). Variants of G2 Theory. A number of variants of G 2 theory have been proposed. The purpose of some of these has been to reduce the computational expense of the calculations while others have been aimed at improving the accuracy. G2(MP2) Theory. A variation of G 2 theory which uses reduced orders of Moller-Plesset perturbation theory is G2(MP2) theory (9). In this theory the basis set extension corrections of G 2 theory in steps 4(a), 4(b), and 4(d) are replaced by a single correction obtained at the M P 2 level with the 6-311+G(3df,2p) basis set, AE'(+3df,2p), as given by step 4'(d) in Table I. The total G 2 ( M P 2 ) energy is thus given by E [ G 2 ( M P 2 ) ] = E[MP4/6-311 G(d,p)] + AE(QCI) + AE'(+3df ,2p) + A E ( H L C ) 0

(4)

+ AE(ZPE) where the AE(QCI) and A E ( H L C ) terms are the same as in G 2 theory. The G2(MP2) energy requires only two single-point energy calculations, QCISD(T)/6-311G(d,p) and MP2/6-311+G(3df,2p), since the sum of the E [ M P 4 / 6 31 lG(d,p)] and AE(QCI) terms in equation (4) is equivalent to the QCISD(T)/6311G(d,p) energy and the QCISD(T)/6-31 lG(d,p) calculation provides the M P 2 / 6 311G(d,p) energy needed to evaluate AE'(+3df,2p). The absence of the M P 4 / 6 311G(2df,p) calculation in G 2 ( M P 2 ) theory provides significant savings in computational time and disk storage such that larger systems can be handled than in G 2 theory. The limiting calculation in G2(MP2) theory is the QCISD(T)/6-31 lG(d,p) calculation. G2(MP2) theory is somewhat less accurate than G 2 theory, having an average absolute deviation of 1.58 kcal/mol for the 125 reaction energies used for validation of G 2 theory (see Table II). G2(MP2,SVP) Theory. A variation of G 2 theory which uses reduced orders of Moller-Plesset perturbation theory in combination with a smaller basis set for the quadratic configuration correction is G 2 ( M P 2 , S V P ) theory (10,11). The " S V P " refers to the split-valence plus polarization basis, 6-31G(d), used in this Q C I S D ( T ) correction. In this theory the final energy is given by M

E [ G 2 ( M P 2 , S VP)] = E[MP4/6-31 G(d)] + AE"(QCI) + AE (+3df,2p) 0

(5)

+ A E ( H L C ) 4- A E ( Z P E ) where the AE"(QCI) and AE"(+3df,2p) correction terms are relative to the 6-31G(d) basis set (see Table I). The H L C for G 2 ( M P 2 , S V P ) theory is A = 5.13 mhartrees and


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B = 0.19 mhartrees. The G 2 ( M P 2 , S V P ) energy requires only two single-point energy calculations, QCISD(T)/6-31G(d) and MP2/6-311+G(3df,2p), since the sum of the E[MP4/6-31G(d)] and AE'"(QCI) terms in equation (5) is equivalent to the QCISD(T)/6-31G(d) energy and the QCISD(T)/6-31G(d) calculation provides the MP2/6-31G(d) energy needed to evaluate AE (+3df,2p). The use of the 6-31G(d) basis set in the quadratic configuration interaction calculation instead of the 631 l G ( d ) basis, as in the G 2 and G2(MP2) theories, reduces computational time and disk space. G 2 ( M P 2 , S V P ) theory has an average absolute deviation of 1.63 kcal/mol for the 125 reaction energies used for validation of G 2 theory (see Table II). This is similar to that of G 2 ( M P 2 ) theory, but the method requires significantly less computer resources. The limiting calculation in G 2 ( M P 2 , S V P ) theory is the QCISD(T)/631G(d) calculation. A comparison of the different G 2 theories is shown in F i g . 1. With the exception of the proton affinities, G 2 theory performs the best of the three methods for the different types of reactions.


M

G 2 ( C O M P L E T E ) Theory. A variation of G 2 theory in which the additivity assumptions are eliminated is referred to as G 2 ( C O M P L E T E ) theory (7). In this method the energy is calculated with the full 6-311+G(3df,2p) basis set using the quadratic configuration method: E [ G 2 ( C O M P L E T E ) ] = E[QCISD(T)/6-311 +G(3df ,2p)] + A E ( H L C ) 0

(6)

+ AE(ZPE) The H L C for G 2 ( C O M P L E T E ) theory is A = 5.13 mhartrees and B = 0.19 mhartrees. Since the method uses the full basis set at the highest correlation level, it is applicable to only small molecules. The accuracy of G 2 ( C O M P L E T E ) is only slightly improved (average absolute deviation of 1.17 kcal/mol for the 125 molecule G 2 test set) over that of G 2 theory. Improvement in Correlation Treatment, Geometries, and Zero-Point Energies. Other modifications of G 2 theory have been investigated which use higher levels of theory for correlation effects, geometries, and zero-point energies (72). A higher level of correlation treatment was examined using Brueckner doubles [BD(T)] (13,14) and coupled cluster [CCSD(T)] (15-17) methods rather than quadratic configuration interaction [QCISD(T)]. These methods are referred to as G 2 ( B D ) and G 2 ( C C S D ) , respectively. The use of geometries optimized at the Q C I S D level rather than the second-order Moller-Plesset level (MP2) and the use of scaled M P 2 zeropoint energies rather than scaled Hartree-Fock (HF) zero-point energies have also been examined. These methods are referred to as G 2 / / Q C I and G 2 ( Z P E = M P 2 ) , respectively (12). The set of 125 reaction energies employed for validation of G 2 theory was used to test these variations of G 2 theory. Inclusion of higher levels of correlation treatment has little effect except in the cases of multiply-bonded systems. In these cases better agreement is obtained in some cases and poorer agreement in others so that there is no improvement in overall performance. The use of Q C I S D geometries yields significantly better agreement with experiment for several cases



182


rrr^

G2 G2(MP2) G2(MP2,SVP)

Figure 1. Average absolute deviations of atomization energies (AEs), ionization potentials (IPs), electron affinities (EAs), and proton affinities (PAs) calculated from G 2 , G 2 ( M P 2 ) , and G 2 ( M P 2 , S V P ) theories for the 125 reaction energies in the G 2 test set.


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including the ionization potentials of C S and O2, electron affinity of C N , and dissociation energies of N2, O2, C N , and SO2. This leads to a slightly better agreement with experiment overall. The use of M P 2 zero-point energies gives no overall improvement. These methods may be useful for specific systems. Other Variants. Bauschlicher and Partridge (18) have proposed a modification of G 2 theory which uses geometries and vibrational frequencies from density functional theory (DFT) methods. In this method, referred to as G 2 ( B 3 L Y P / M P 2 / C C ) , the QCISD(T) step in G 2 ( M P 2 ) theory is replaced by a C C S D ( T ) calculation. In addition, the HF/6-31G(d) zero-point energies and M P 2 / 6 31G(d) geometries are replaced by zero-point energies and geometries obtained from density functional theory [B3LYP/6-31G(d)]. This modifcation does not improve the average absolute deviation, but does decrease the maximum errors compared with the G 2 ( M P 2 ) approach. Mebel, Morokuma and L i n (19,20) have proposed a family of modified G 2 schemes based on geometry optimization and vibrational frequency calculations using density functional theory [B3LYP/6-31 lG(d,p)] and electron correlation using coupled cluster methods. These schemes use spin-projected Moller-Plesset theory for radicals and triplets and may be more reliable for radicals with large spin contamination. Extension to Third-Row Non-Transition Metal Elements. G 2 theory has been extended to include molecules containing third-row non-transition elements G a - K r (27). Basis sets compatible with those used in G 2 theory for first- and second-row molecules were derived for this extension. G 2 theory for the third-row incorporates the following modifications: 1. The M P 2 geometry optimizations and the H F vibrational frequency calculations use the 641(d) basis set of Binning and Curtiss (22) for G a - K r along with 631G(d) for first- and second-row atoms, referred to overall for simplicity as "63 l G ( d ) . " The same scale factor (0.8929) is used for the zero-point energies. 2.

The M P 2 , M P 4 and QCISD(T) calculations (step 4 in Section 2.2.1) use the 6-311G basis and appropriate supplementary functions for first- and second-row atoms, and corresponding sets that were developed (27) for G a - K r , referred to overall again for simplicity as "6-311G."

3.

The splitting factor of the d-polarization functions for the 3df basis set extension is three rather than the factor of four used for first- and second-row atoms. The 3d core orbitals and Is virtual orbitals are frozen in the single-point correlation calculations.

4. First-order spin-orbit energy corrections, AE(SO), are included in the G 2 energies for the third-row species that have first-order spin-orbit effects. This includes P and P atoms and n molecules. Values for these corrections are obtained from spin-orbit configuration interaction calculations (23). 2

3

2


184



The average absolute deviation from experiment of 40 test cases involving species containing G a - K r atoms (atomization energies, ionization energies, electron affinities, proton affinities) is 1.37 kcal/mol. This is only slightly greater than for the G 2 treatment of first- and second-row molecules for the 125 reaction energies of small molecules, for which the average absolute deviation is 1.21 kcal/mol. The inclusion of first-order atomic and molecular spin-orbit corrections is important for attaining good agreeement with experiment. When the spin-orbit correction is not included, the deviation increases to 2.36 kcal/mol. The G 2 ( M P 2 ) and G 2 ( M P 2 , S V P ) theories can also be used for the third-row. The formulation is analogous to the first- and second-rows with the exception of the additional spin-orbit correction term. The average absolute deviation for the third-row test set using G 2 ( M P 2 ) and G 2 ( M P 2 , S V P ) theories is slightly larger (1.92 kcal/mol for both methods). Density Functional Methods. Over the past 30 years, density functional theory has been widely used by physicists to study the electronic structure of solids. M o r e recently chemists have been using the Kohn-Sham version (24) of density funtional theory ( D F T ) as a cost-effective method to study properties of molecular systems. The density functional models currently being used by quantum chemists may be broadly divided into non-empirical and empirical types. The simplest non-empirical type is the local spin density functional, which treats the electronic environment of a given position in a molecule as if it were a uniform gas of the same density at that point. One of these is the S V W N functional that uses the Slater functional (25) for exchange and the uniform gas approximate correlation functional of Vosko, W i l k and Nusair (26). The more sophisticated functional B P W 9 1 combines the 1988 exchange functional of Becke (27) with the correlation functional of Perdew and Wang (28). Both components involve local density gradients as well as densities. The Becke part involves a single parameter which fits the exchange functional to accurate computed atomic data. The B P 8 6 functional is similar but uses the correlation functional of Perdew (29). The B L Y P (30) functional also uses the Becke 1988 part for exchange, together with the correlation part of Lee, Yang and Parr (31). This L Y P functional is based on a treatment of the helium atom and really only treats correlation between electrons of opposite spin. A number of other functionals use parameters which are fitted to energies in the original G 2 test set. These give a functional which is a linear combination of Hartree-Fock exchange, 1988 Becke exchange, and various correlation parts. This idea was introduced by Becke (32) who obtained parameters by fitting to the molecular data. This is the basis of the B 3 P W 9 1 functional. The others (B3P86 and B 3 L Y P ) are constructed in a similar manner, although the parameters are the same as in B 3 P W 9 1 . In several validation studies for molecular geometries and frequencies, D F T has given results of quality similar to that of M P 2 theory (33,34). It has also been examined for use in calculation of thermochemical data (32-38). For example, Becke (32) found that the B 3 P W 9 1 functional with a numerical basis set gave an average absolute deviation of 2.4 kcal/mol for the 55 atomization energies in the original G 2



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test set, about twice as large as G 2 theory. He also found similar results for ionization potentials and proton affinities. Bauschlicher (37) has examined several D F T methods [ B L Y P , B 3 L Y P , BP86, B3P86, B P ] for the 55 atomization energies in the original G 2 test set using the same 6-311+G(3df,2p) basis set. He found that B 3 L Y P gave the best agreement with experiment (average absolute deviation of 2.20 kcal/mol). The results for the 125 reaction energies in the original G 2 test set are given in Table E for the seven density functional ( S V W N , B L Y P , B P W 9 1 , BP86, B 3 L Y P , B 3 P W 9 1 , B3P86) described above. The results are based on the 6-311+G(3df,2p) basis set and the geometries and zero-point energies used in the original G 2 paper (7). The local density approximation has a very large average absolute deviation (26.2 kcal/mol) especially for the atomization energies due to overbinding. The gradient corrected functional have average absolute deviations of 10 kcal/mol or less. The methods containing the Becke three-parameter functional perform better than the Becke 1988 functional for atomization energies, ionization energies, and proton affinities. For electron affinities the smallest average absolute deviation is found for the B P W 9 1 functional. Overall, the B 3 L Y P functional performs the best, with an average absolute deviation of 2.81 kcal/mol. The B3PW91 functional is only slightly worse with an average absolute deviation of 2.89 kcal/mol. Assessment of Theoretical Methods for Computational Thermochemistry Critical documentation and evaluation of theoretical models for calculating energies are essential if such methods are to become proper tools for chemical investigation. A s mentioned earlier, G 2 theory was tested on a total of 125 reaction energies, chosen because they have well-established experimental values (7). A l l of the molecules contained only one or two non-hydrogen atoms with two exceptions (CO2 and SO2). In recent work (39), the test set has been expanded to include larger, more diverse molecules with enthalpies of formation at 298 K being used for comparison between experiment and theory. This set, referred to as the "enlarged G 2 neutral test set," includes the 55 molecules whose atomization energies were used to test G 2 theory and 93 new molecules. The full set includes 29 radicals, 35 non-hydrogen systems, 22 hydrocarbons, 47 substituted hydrocarbons, and 15 inorganic hydrides. The set includes molecules containing up to six non-hydrogen atoms. In this section we critically evaluate G 2 theory, some of its variants, and several density functional methods on this enlarged G 2 test set. Before doing this, we describe the procedure employed to calculate enthalpies of formations from calculated energies. Calculation of Enthalpies of Formation. One of the methods for calculating enthalpies of formation at 0 K is based on subtracting calculated atomization energies ID from known enthalpies of formation of the isolated atoms. For any molecule, such as A B H , the enthalpy of formation at 0 K is given by 0

x

y

z

A H° ( A B H , 0 K ) = xAfH°(A, 0 K ) + yA H° (B, 0 K ) + zA H° (H, 0 K ) f

x

y

z

f

-£D

f

0


(7)

186


Experimental values (40) for the atomic AfH° are used. The experimental enthalpies of formation of S i , Be, and A l have large uncertainties (2.0, 1.2 and 1.0 kcal/mol, respectively). This means that the calculated enthalpies of formation containing these atoms w i l l have uncertainties due to the use of the atomic enthalpies in E q . (7) and the theoretical methods. The other atomic enthalpies for first- and second-row elements are quite accurate (±0.2 kcal/mol or better). Theoretical enthalpies of formation at 298 K are calculated by correction to A H ° ( 0 K ) as follows: Downloaded by NORTH CAROLINA STATE UNIV on September 9, 2013 | http://pubs.acs.org Publication Date: February 1, 1998 | doi: 10.1021/bk-1998-0677.ch010

f

AfH°(A B H , 298 K ) = A H° ( A B H , 0 K ) + x

y

z

f

x

y

(8)

z

[ H ° ( A B H , 298 K ) - H ° ( A B H , 0 K)] - x[H°(A, 298 K ) - H°(A, 0 K ) ] - y[H°(B, 298 K ) - H°(B, 0 K ) ] - z[H°(H, 298 K ) - H°(H, 0 K ) ] x

y

z

x

y

z

st

st

st

The enthalpy corrections (in square brackets) are treated differently for compounds and elements. The correction for the A B H molecule is made using scaled H F / 6 31G(d) frequencies for the vibrations in the harmonic approximation for vibrational energy (41), the classical approximation for translation (3RT/2) and rotation (3RT/2 for nonlinear molecules, R T for linear molecules) and the P V term. The harmonic approximation may not be appropriate for some low frequency torsional modes, although the error should be small in most cases. The elemental corrections are for the standard states of the elements [denoted as "st" in eqn (8)] and are taken directly from experimental compilations (40). The resulting values of A H° (298 K ) are often discussed as theoretical numbers, although they are based on some experimental data for monatomic and standard species. Enthalpies of formation of cations are calculated by combining enthalpies of formation and ionization potentials of the corresponding neutrals (42). x

y

z

f

Gaussian-2 Theory and Variants. The enlarged G 2 neutral test set of 148 molecules was used to assess the performance of G 2 , G2(MP2) and G 2 ( M P 2 , S V P ) theories (39). The mean absolute deviation between the theoretical and experimental enthalpies of formation at 298 K for the different methods is listed in Table HI. The results indicate that G 2 theory is the most reliable of the methods, with a mean absolute deviation of 1.58 kcal/mol for the 148 enthalpies. This is larger than for the atomization energies of the 55 small molecules in the original G 2 test set, mainly due to the new molecules containing multiple halogens and molecules with unsaturated rings. The largest deviations between experiment and G 2 theory (up to 8 kcal/mol) occur for molecules having multiple halogens. This leads to a poor performance of G 2 theory on nonhydrogen systems (see Table HI). The G 2 enthalpies of formation for cyclic hydrocarbons with unsaturated rings deviate with experiment by 2-4 kcal/mol. The other hydrocarbons are generally in good agreement with experiment. The modified versions of G 2 theory, G 2 ( M P 2 ) and G 2 ( M P 2 , S V P ) , have average absolute deviations of 2.04 and 1.93 kcal/mol, respectively. Both methods do poorly for non-hydrogens similar to G 2 theory. G 2 ( M P 2 , S V P ) theory appears to be


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very good for hydrocarbons, radicals, and inorganic hydrides. Surprisingly, this approximation does better for hydrocarbons than G 2 theory, especially cyclic systems, for which it has an average absolute deviation of 1.06 kcal/mol. Since G 2 ( M P 2 , S V P ) theory uses considerably less cpu time and disk storage than G 2 theory, it may be a useful alternative for large hydrocarbons. Table III. Average absolute deviation (in kcal/mol) between theory and experiment for


148 enthalpies of formation in the enlarged G 2 neutral test set

a

Type of Moleculeb Method D All C A B G2 1.16 1.48 1.58(8.2) 1.29 2.53 G2(MP2) 1.36 2.04(10.1) 1.83 1.89 3.30 G2(MP2,SVP) 2.04 1.20 1.93(12.5) 0.77 3.57 SVWN 54.64 124.40 90.88(228.7) 133.71 73.65 BLYP 7.09(28.4) 6.10 6.09 8.09 10.30 BPW91 6.48 7.85(32.2) 7.99 4.85 12.25 BP86 15.76 25.82 26.80 20.19(49.7) 16.61 B3LYP 2.98 3.11(20.1) 2.76 2.10 5.35 3.21 B3PW91 2.77 3.51(21.8) 3.96 5.14 B3P86 13.53 17.97(49.2) 30.81 25.49 7.80 From Ref. 39. A = non-hydrogen systems (35); B = hydrocarbons (22); C = substituted hydrocarbons (47); D = radicals (29); E = inorganic hydrides (15). Number of molecules in each type is listed in parentheses. Maximum absolute deviation in parentheses. c

a

b

E 0.95 1.20 0.91 33.80 3.13 4.21 8.16 1.84 1.99 7.86

Gaussian-2 theory is based on non-relativistic energies, but for large molecules, such as those in the enlarged G 2 neutral test set, spin-orbit effects become important in some cases. T o take account of these effects a spin-orbit correction A E ( S O ) can be added to the G 2 energy: E [ G 2 ] = E [ G 2 ] + AE(SO) 0

S 0

(9)

0

For first- and second-row molecules the spin-orbit effect is significant for the cases when it is a first-order effect, such as in P and P atoms and n molecules. It can be neglected for the other atoms and molecules of the first- and second-row for which it is not a first-order effect. The spin-orbit corrections can be derived from experimental data in Moore's tables (43) or, alternatively, they can be calculated quite accurately (23,44). The corrections are large enough for heavier atoms such as CI (-1.34 mhartree per atom), S (-0.89 mhartree per atom) and F (-0.61 mhartree per atom) that in molecules containing several such atoms the effect on the enthalpy of formation w i l l amount to several kcal/mol. For the 148 enthalpies of formation in the G 2 neutral test set inclusion of spin-orbit effects reduces the average absolute deviation to 1.47 kcal/mol from 1.58 kcal/mol without spin-orbit corrections (39). The spin-orbit correction significantly improves the enthalpies for the chlorine substituted 2

3

2



188

molecules, but little overall improvement is seen for the fluorine containing


molecules.

Density Functional Theory Thermochemistry. The seven D F T methods described in the theory section have been assessed on the enlarged G 2 test set (39). The D F T calculations were done with the 6-311+G(3df,2p) basis set and the geometries and zero-point energies used in the original G 2 paper (7). The average absolute deviations for the G 2 and D F T methods are illustrated in F i g . 2 and broken up into types of molecules in Table HI. The D F T methods give a wide range of average absolute deviations (3.11 to 90.9 kcal/mol) for the G 2 test set, all of which are larger than for the G 2 methods. A s expected, the local density method ( S V W N ) performs poorly with a deviation of 90.9 kcal/mol and overbinds all systems except L i . However, this method involves no parameterization, and application of empirical corrections as in other methods can significantly improve its performance. For the remaining gradient corrected functional, the average absolute deviation ranges from 3.11 to 20.19 kcal/mol. The Becke three parameter functional performs better than the Becke exchange functional with all three correlation functionals. The B 3 L Y P functional performs the best of the functionals tested with an average absolute deviation of 3.11 kcal/mol. The deviation for the B 3 P W 9 1 functional is only slightly larger at 3.51 kcal/mol. A s in the case of the G 2 methods, the D F T methods do poorest for the systems containing multiple halogens as is seen from the large average absolute deviations for the non-hydrogen systems in Table III. 2

The maximum deviations of the D F T methods are significantly larger than those of the G 2 methods (see Table EI). For example, B 3 L Y P has a maximum deviation of 20.1 kcal/mol compared to 8.2 kcal/mol for G 2 theory. The largest errors for the B 3 L Y P method occur for non-hydrogen systems (average absolute deviation of 5.35 kcal/mol) while hydrocarbons, substituted hydrocarbons, and radicals have smaller average absolute deviations (2 to 3 kcal/mol). The distribution of deviations for B 3 L Y P is given in F i g . 3. About 50% of the B 3 L Y P enthalpies fall within ± 2 kcal/mol of the experimental values and 63% fall within ± 3 kcal/mol. In comparison, over 70% of the G 2 enthalpies fall within ± 2 kcal/mol of the experimental values and 87% fall within ± 3 kcal/mol. While the deviations for G 2 theory are quite equally distributed (Fig. 3), the B 3 L Y P method has more negative deviations (underbinding). The B 3 L Y P distribution covers a much larger range (-20 to 8 kcal/mol) than G 2 theory (-7 to 8 kcal/mol). The performance measures discussed above have important consequences. The best performing B 3 L Y P functional has an average absolute deviation (3.11 kcal/mol) almost twice that of G 2 theory. Among the 148 molecules studied, only 5 have deviations of 5 kcal/mol or more with G 2 theory, whereas 25 molecules have deviations of more than 5 kcal/mol with the B 3 L Y P functional. These considerations may be important for assessing the thermochemistry of systems where there is disagreement between theory and experiment or for making predictions for systems where there are no experimental measurements. Typical timings of several D F T calculations for comparison with the G 2 methods are shown for two molecules in



10.



189

Figure 2. Average absolute deviations of G 2 and D F T methods for the 148 molecule enlarged G 2 test set of 148 enthalpies. (The S V W N density functional method has an average absolute deviation of 90.88 kcal/mol).



190


D e v i a t i o n (Expt - G2), k c a l / m o l

D e v i a t i o n (Expt-B3LYP), k c a l / m o l

Figure 3. Histogram of G 2 and B 3 L Y P deviations for the enlarged G 2 test set of 148 molecules. Each vertical bar represents one kcal/mol range for the G 2 results and two kcal/mol range for the B 3 L Y P results. (Reprinted with permission from Ref. 39. Copyright 1997 American Institute of Physics.)


10.



191

Table I V . The D F T methods are about 80 times faster than G 2 theory and 6 times faster than G 2 ( M P 2 , S V P ) theory for these molecules.

Table IV. Comparison of cpu times and disk storage used in G2 and D F T calculations on benzene and t-butyl radical a

Benzene ( D ) disk storage

t-Butyl Radical (C ) cpu time disk storage

6h


Method

cpu time

G2 G2(MP2) G2(MP2,SVP) BLYP BPW91 B3LYP B3PW91 Using Gaussian 94 storage in G b . a

3v

851 1.4 868 1.1 170 0.6 226 0.6 58 0.4 64 0.4 10 0.1 7 0.1 6 0.1 10 0.1 10 0.1 10 0.1 11 0.1 9 0.1 (Ref. 8) on a Cray Y M P - C 9 0 . Time in minutes and maximum

In the assessment of the D F T methods on the enlarged G 2 test set MP2(full)/6-31G(d) optimized geometries and scaled HF/6-31G(d) vibrational frequencies were employed (39). The D F T geometries and zero-point energies for thermochemical calculations may be useful and have been examined in several studies (37,45-47). The results have indicated that D F T geometries and zero-point energies generally perform as well as those from ab initio methods as long as appropriate scale factors for the zero-point energies are used. Applications Gaussian-2 theory has been applied to many molecular systems and has in most cases been quite successful. It has been used to make predictions of bond dissociation energies, ionization energies, electron affinities, appearance energies, proton affinities, and enthalpies of formation. There have been several reviews of these applications (48,49). In this article we describe several typical examples of the use of G 2 theory to obtain thermochemical data. Atomization Energies. A s discussed in the previous sections G 2 theory was originally tested on 55 atomization energies (average absolute deviation of 1.19 kcal/mol) and in subsequent work it was tested on 148 enthalpies of formation at 298 K (average absolute deviation of 1.48 kcal/mol). It has been applied to numerous molecules to determine bond energies and enthalpies of formation. A n example of this is the application to C H and S i H hydrides (1,42). For acetylene the experimental D ( H C C - H ) values have fallen in two groups, one around 126 kcal/mol and the other around 131-132 kcal/mol. The G 2 value of 133.4 kcal/mol is in agreement with the larger experimental value. In a recent review of the experimental 2

n

2

n

C


192


data, Berkowitz, Ellison and Gutman (50) give a value of 131.3 ± 0.7 kcal/mol, in agreement with the G 2 results. For ethylene the experimental D ( H C C H - H ) values have fallen in the range 98-117 kcal/mol. The G 2 value is 110.1 kcal/mol. In their review of the experimental data, Berkowitz, Ellison and Gutman (50) give a value of 109.7 ± 0.7 kcal/mol, in agreement with the G 2 results.


0

2

Ionization Energies. G 2 theory was originally tested on 38 ionization potentials and found to be in good agreement with experiment (average absolute deviation of 1.24 kcal/mol). It has been used to calculate adiabatic ionization energies of numerous molecules. Some of these calculations (1,42,51-53) are summarized in Table V . The G 2 results for the adiabatic ionization energies of S i H hydrides (42) n = 2, 4-6 are in agreement with measurements reported by Ruscic and Berkowitz (54,55) from a photoionization study. The G 2 adiabatic ionization potential (57) of C H 3 O (10.78 e V ) is in agreement with the value of 10.726 ± 0.008 e V reported by Ruscic and Berkowitz (56) from a photoionization study. A previous photoelectron value (57) of 7.37 ± 0.03 e V appears to be incorrect. In a separate study (52) the ionization energy of C H 0 was calculated by G 2 theory and is in good agreement with experiment (see Table V ) . 2

2

n

5

Electron Affinities. G 2 theory was originally tested on 25 electron affinities and found to be in good agreement with experiment (average absolute deviation of 1.29 kcal/mol). It has since been used to calculate electron affinities of other molecules and clusters. For example, it was used to calculate the electron affinities of silicon clusters (Si ) containing up to five silicon atoms (58). The theoretical electron affinities are in good agreement with the experimental data on n = 1-4. n

Table V . Adiabatic ionization potentials (in eV) of selected molecules Expt. G2 Theory C H 3 O -> C H 0

+

C H 0 -> C H 0

+

CH SH-> CH SH

+

2

5

2

5

3

+

2

2

-> C H

2

C H

4

-> C H

4

2

2

+

S i H -> S i H 2

2

2

S i H -> S i H 2

4

2

S i H -> S i H 2

a

b

c

d

e

f

b

10.726

+e

10.32

c

10.29

+e

9.46

3

C H 2

10.78

+ e"

3

6

2

+ 2 + 4 + 6

9.44

d

+e

11.42

e

11.40

+ e"

10.32

6

10.29

+ e"

8.30

+ e"

8.11

+ e"

f

8.20

f

9.74

f

9.70

8.09

References to experiment given in the theory references. R e f . 51. Ref. 52. Ref. 53. R e f . 1. R e f . 42.



10.



193

Proton Affinities and Gas Phase Acidities. G 2 theory was originally tested on seven proton affinities and found to be in good agreement with experiment (average absolute deviation of 1.04 kcal/mol). Smith and Radom (59) have used G 2 theory to obtain proton affinities for 31 small molecules. The theoretical values are in good agreement for a range of bases with proton affinities spanning some 120 kcal/mol. Smith and Radom found a small number of discrepant cases, which can be reconciled if the currently accepted proton affinity of isobutene is lowered by 2-5 kcal/mol. Smith and Radom (60) also evaluated G 2 theory in the calculation of gas phase acidities of 23 molecules. They found it to perform very well, with a mean error of 1.24 kcal/mol and consistently within 2 kcal/mol of reliable experimental data. In a small number of cases they found a larger discrepancy between experiment and theory and suggested that in these cases a re-examination of the experimental data may be warranted. Conclusions Gaussian-2 theory is a technique based on ab initio molecular orbital theory that has been widely used for the calculation of accurate thermochemical data. It is a composite method which is based on a well-defined sequence of calculations and has been tested on molecules having well-established experimental data. The original G 2 test set contains 125 reaction energies including atomization energies, ionization potentials, electron affinities, and proton affinities. The average absolute deviation of G 2 theory from experiment for this test set was 1.21 kcal/mol. In recent work, the test set has been expanded to include larger, more diverse molecules with comparison between theory and experiment being made using enthalpies of formation at 298 K . This set, referred to as the "enlarged G 2 neutral test set," includes 148 molecules. The average absolute deviation of G 2 theory from experiment for this new test set is 1.58 kcal/mol, slightly larger than the smaller test set. The largest deviations between experiment and G 2 theory occur for molecules having multiple halogens and molecules having unsaturated rings. A number of variants of G 2 theory such as G 2 ( M P 2 ) and G 2 ( M P 2 , S V P ) theory are available which save considerable computational time with some loss in accuracy. Gaussian-2 theory is the most accurate of the G 2 methods, while G2(MP2) and G 2 ( M P 2 , S V P ) are cost-effective alternatives. In addition, Petersson et al. (61-63) have developed a series of related methods, referred to as complete basis set ( C B S ) methods, that are reviewed elsewhere in this book. The other type of method that we have described in this chapter is density functional theory. These methods require much less computational resources than the G 2 methods, but are not as accurate. The D F T methods have a wide range of average absolute deviations (3.11 to 90.9 kcal/mol) for the enthalpies in the enlarged G 2 test set, all of which are larger than for the G 2 methods. The best performing B 3 L Y P functional has an average absolute deviation (3.11 kcal/mol) almost twice that of G 2 theory and a much larger distribution of errors. The B 3 P W 9 1 functional has an average absolute deviation of 3.51 kcal/mol, similar to that of B 3 L Y P . The results of the assessment on the enlarged G 2 test set indicate that B 3 L Y P with a large basis set



194

such as 6-311+G(3df,2p) is the preferred D F T method for thermochemical calculations at this time. The accuracy of the D F T methods can be significantly improved when used in schemes based on isodesmic bond separation reaction energies (64). Density functional theory is also useful for calculating geometries and vibrational frequencies for thermochemical calculations.


Acknowledgment This work was supported by the U . S . Department of Energy, Division of Materials Sciences, under contract No. W-31-109-ENG-38. Literature Cited 1. Curtiss, L. A.; Raghavachari, K.; Trucks, G.W.; Pople, J. A . J. Chem. Phys. 1991, 94, 7221. 2. Pople J.A.; Head-Gordon, M . ; Fox, D.J.; Raghavachari, K.; Curtiss, L.A. J. Chem. Phys. 1989, 90, 5622. 3. Curtiss, L.A.; Jones, C.; Trucks, G. W.; Raghavachari, K.; Pople, J. A . J. Chem. Phys. 1990 93, 2537. 4. Hehre, W. J.; Radom, L . ; Pople, J.A.; Schleyer, P.v.R. Ab Initio Molecular Orbital Theory; John Wiley: New York, 1987. 5. Pople, J. A.; Schlegel, H. B.; Krishnan, R.; Defrees, D. J.; Binkley, J. S.; Frisch, M . J.; Whiteside, R. A.; Hout, R. F.; Hehre, W.J. Int. J. Quantum Chem. Symp. 1981, 15, 269. 6. Pople, J. A.; Head-Gordon, M.; Raghavachari, K. J. Chem. Phys. 1987, 87, 5968. 7. Curtiss, L. A.; Carpenter, J. E.; Raghavachari, K.; Pople J. A. J. Chem. Phys. 1992, 96, 9030. 8. Frisch, M . J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M . W.; Johnson, B. G . ; Robb, M . A.; Cheeseman, J. R.; Keith, T. A.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M . A.; Zakrzewski, V. G.; Ortiz, J. V . ; Foresman, J. B.; Cioslowski J.; Stefanov, B. B.; Nanayakkara, A.; Challacombe, M . ; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M . W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; Head-Gordon, M . ; Gonzales, C.; Pople, J. A.; Gaussian 94, Gaussian, Inc. Pittsburgh, PA, 1995. 9. Curtiss, L . A.; Raghavachari, K.; Pople, J. A. J. Chem. Phys. 1993, 98, 1293. 10. Smith, B. J.; Radom, L. J. Phys. Chem. 1995, 99, 6468. 11. Curtiss, L. A.; Redfern, P. C.; Smith, B. J.; Radom, L. J. Chem. Phys. 1996, 104, 5148. 12. Curtiss, L . A.; Raghavachari, K.; Pople, J. A. J. Chem. Phys. 1995, 103, 4192. 13. Handy, N. C.; Pople, J. A.; Head-Gordon, M . ; Raghavachari, K.; Trucks, G. W. Chem. Phys. Lett. 1989, 164, 185. 14. Raghavachari, K.; Pople, J. A.; Replogle, E. S.; Head-Gordon, M . J. Phys. Chem. 1990, 94, 5579. 15. Cizek, J. J. Chem. Phys. 1966, 45, 4256.


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